Automorphisms of countable recursively saturated models

Extending automorphisms

Let $G$ be the automorphism group of a countable recursively saturated model of $PA$. For every nontrivial $f\in G$ there is a countable recursively saturated elementary end extension $N$ such that $f$ has no extension to an automorphism of $N$. It is open whether there is an $f\in G$ such that for all countable recursively saturated elementary end extensions $N$, $f$ does not extend to an automorphism of $N$?

If $S$ is an inductive partial satisfaction class of $M$, and $f\in{\rm Aut}(M,S)$, then there is a countable recursively saturated elementary end extension $N$ such that $f$ extends to an automorphism of $N$. If $M$ is arithmetically saturated, then there are $f\in G$ such that for every inductive partial satisfaction class $S$ of $M$, $f\notin {\rm Aut}(M,S)$. Problem: what if $M$ is recursively, but not arithmetically, saturated?